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表示行内公式
$$ $$

表示公式块


平方差公式


$ a2-b2= (a+b)(a-b) $
$ a^2-b^2= (a+b)(a-b) $



立方差公式


$ a3-b3 = (a+b)(a2-ab+b2) $
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                        数学公式以及对应的 markdown 代码
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                <pre><code class="language-mathematica">$ $
</code></pre>
<p>表示<strong>行内公式</strong></p>
<pre><code class="language-mathematica">$$ $$
</code></pre>
<p>表示<strong>公式块</strong></p>
<ol>
<li>
<h2 id="平方差公式"><a class="header-anchor" href="#平方差公式"></a>平方差公式</h2>
</li>
</ol>
<p>$ a<sup>2-b</sup>2= (a+b)(a-b) $</p>
<pre><code class="language-math">$ a^2-b^2= (a+b)(a-b) $
</code></pre>
<ol start="2">
<li>
<h2 id="立方差公式"><a class="header-anchor" href="#立方差公式"></a>立方差公式</h2>
</li>
</ol>
<p>$ a<sup>3-b</sup>3 = (a+b)(a<sup>2-ab+b</sup>2) $</p>
<pre><code class="language-math">$ a^3-b^3 = (a+b)(a^2-ab+b^2) $
</code></pre>
<ol start="3">
<li>
<h2 id="立方和公式"><a class="header-anchor" href="#立方和公式"></a>立方和公式</h2>
</li>
</ol>
<p>$ a<sup>3+b</sup>3 = (a-b)(a<sup>2+ab+b</sup>2)$</p>
<pre><code class="language-mathematica">$ a^3+b^3 = (a-b)(a^2+ab+b^2)$
</code></pre>
<ol start="4">
<li>
<h2 id="完全平方公式"><a class="header-anchor" href="#完全平方公式"></a>完全平方公式</h2>
</li>
</ol>
<p>$(a + b)^2  = a^2 + 2ab+b^2 ​$<br>
$(a - b)^2  = a^2 - 2ab+b^2 ​$</p>
<pre><code class="language-mathematica">$(a + b)^2  = a^2 + 2ab+b^2 $
$(a - b)^2  = a^2 - 2ab+b^2 $
</code></pre>
<ol start="5">
<li>
<h2 id="一元二次方程求根公式"><a class="header-anchor" href="#一元二次方程求根公式"></a>一元二次方程求根公式</h2>
</li>
</ol>
<p>$ ax^2+bx+c =0$<br>
$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$</p>
<pre><code class="language-mathematica">$ ax^2+bx+c =0$
$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
</code></pre>
<ol start="6">
<li>
<h2 id="韦达定理"><a class="header-anchor" href="#韦达定理"></a>韦达定理</h2>
<p>设$x1,x2​$是一元二次方程$ ax^2+bx+c =0​$的两个根，则$x1,x2​$满足:</p>
<p><em>$x1+x2 = -\frac{b}{a}​$</em><br>
<em>$ x1 \cdot x2 = \frac{c}{a}​$</em></p>
<pre><code class="language-mathematica">*$x1+x2 = -\frac{b}{a}$*
*$ x1 \cdot x2 = \frac{c}{a}$*
</code></pre>
</li>
<li>
<h2 id="有关集合的公式"><a class="header-anchor" href="#有关集合的公式"></a>有关集合的公式</h2>
<p>设$I$为全集，$\varnothing$为空集，如果$A$是$I$的子集,$B$是$I$的子集，则$A \subset I $, $ B \subset I $。<br>
于是，</p>
<p><em>$I \cup A = I​$</em><br>
<em>$I \cap \varnothing = \varnothing​$</em></p>
<pre><code class="language-mathematica">*$I \cup A = I$*
*$I \cap \varnothing = \varnothing$*
</code></pre>
<p>若$ \bar A ={x|x \in I且x \notin A,A \subseteq I }​$,<br>
　$ \bar B ={x|x \in I且x \notin B,B \subseteq I }​$<br>
则</p>
<p>$ A \cup  \bar A = I$<br>
$ A \cap \bar A = \varnothing$<br>
$\overline {A \cap B} = \bar A \cup \bar B$<br>
$\overline {A \cup B} = \bar A \cap \bar B$</p>
<pre><code>$ A \cup  \bar A = I$
$ A \cap \bar A = \varnothing$
$\overline {A \cap B} = \bar A \cup \bar B$
$\overline {A \cup B} = \bar A \cap \bar B$
</code></pre>
</li>
<li>
<h2 id="不等式"><a class="header-anchor" href="#不等式"></a>不等式</h2>
</li>
</ol>
<p>8.1) $|a| \ge 0​$</p>
<pre><code>```mathematica
$|a| \ge 0$
```
</code></pre>
<p>8.2) $|a|-|b| \le |a+b| \le |a|+|b|$</p>
<pre><code class="language-mathematica">  $|a|-|b| \le |a+b| \le |a|+|b|$
</code></pre>
<p>8.3) $|a| \le b $</p>
<pre><code class="language-mathematica">  $|a| \le b $
</code></pre>
<p>8.5) $|a| \le b \Leftrightarrow -b \le a \le b　 (b&gt;0)$</p>
<pre><code class="language-mathematica">  $|a| \le b \Leftrightarrow -b \le a \le b　 (b&gt;0)$
</code></pre>
<p>8.6) $a<sup>2+b</sup>2 \gt 2ab　(a,b \in R)$</p>
<pre><code class="language-mathematica">  $a^2+b^2 \gt 2ab　(a,b \in R)$
</code></pre>
<p>8.7) $\frac{a+b}{2}  \ge \sqrt{ab}　(a,b \in R^+)$</p>
<pre><code class="language-mathematica">  $\frac{a+b}{2}  \ge \sqrt{ab}　(a,b \in R^+)$
</code></pre>
<p>8.7) $\frac{b}{a} + \frac{a}{b} \ge 2　(ab&gt;0)$</p>
<pre><code class="language-mathematica">  $\frac{b}{a} + \frac{a}{b} \ge 2　(ab&gt;0)$
</code></pre>
<p>8.7) $\frac{a+b+c}{3} \ge \sqrt[3]{abc}　(a,b,c \in R)$</p>
<pre><code class="language-mathematica">  $\frac{a+b+c}{3} \ge \sqrt[3]{abc}　(a,b,c \in R)$
</code></pre>
<p>8.7) $\frac{a_1+a_2+ \ldots +a_n}{n} \ge \sqrt[n]{a_1a_2\ldots a_n}　a_1,a_2, \ldots,a_n \in R^+ n \in N且n&gt;1$</p>
<pre><code class="language-mathematica">$\frac{a_1+a_2+ \ldots +a_n}{n} \ge \sqrt[n]{a_1a_2\ldots a_n}　a_1,a_2, \ldots,a_n \in R^+ n \in N且n&gt;1$
</code></pre>
<ol start="9">
<li>
<h2 id="排列组合"><a class="header-anchor" href="#排列组合"></a>排列组合</h2>
</li>
</ol>
<p>9.1) $ A_n^m = \frac{n!}{(n-m)!} $</p>
<pre><code class="language-mathematica">  $ A_n^m = \frac{n!}{(n-m)!} $
</code></pre>
<p>9.2) $ C_n^m = \frac{n!}{m!(n-m)!} $</p>
<pre><code class="language-mathematica">  $ C_n^m = \frac{n!}{m!(n-m)!} $
</code></pre>
<p>9.3) $ C_n^m = C_n^(n-m)$</p>
<pre><code class="language-mathematica">  $ C_n^m = C_n^(n-m)$
</code></pre>
<p>9.4) $ C_{n+1}^m = C_n^m + C_n^{m-1}$</p>
<pre><code class="language-mathematica">  $ C_{n+1}^m = C_n^m + C_n^{m-1}$
</code></pre>
<p>9.5) $C_n<sup>0+C_n</sup>1+\ldots +C_n^n = 2^n$</p>
<pre><code class="language-mathematica">  $C_n^0+C_n^1+\ldots +C_n^n = 2^n$
</code></pre>
<ol start="10">
<li>
<h2 id="积分公式"><a class="header-anchor" href="#积分公式"></a>积分公式</h2>
</li>
</ol>
<p>10.1) $\int sin(x)dx = cos(x) +C$</p>
<pre><code class="language-mathematica">  $\int sin(x)dx = cos(x) +C$
</code></pre>
<p>10.2) $\int cos(x)dx = -sin(x) +C$</p>
<pre><code class="language-mathematica">  $\int cos(x)dx = -sin(x) +C$
</code></pre>
<p>10.3) $\int tan(x)dx = -ln|cos(x)| +C$</p>
<pre><code class="language-mathematica">  $\int tan(x)dx = -ln|cos(x)| +C$
</code></pre>
<p>10.4) $\int cot(s)dx = ln|sin(x)| +C$</p>
<pre><code class="language-mathematica">  $\int cot(s)dx = ln|sin(x)| +C$
</code></pre>
<p>10.5) $\int sec^2(x)dx = tan(x) +C$</p>
<pre><code class="language-mathematica">  $\int sec^2(x)dx = tan(x) +C$
</code></pre>
<ol start="11">
<li>
<h2 id="导数公式"><a class="header-anchor" href="#导数公式"></a>导数公式</h2>
</li>
</ol>
<p>11.1) $©’=0 $</p>
<pre><code>```mathematica
$(C)'=0 $
```
</code></pre>
<p>11.2) $(sin ,x)’ = cos ,x$<br>
<code>mathematica     $(sin \,x)' = cos \,x$     </code></p>
<p>11.3) $(tan ,x)’ = sec^2 ,x$<br>
<code>mathematica     $(tan \,x)' = sec^2 \,x$     </code></p>
<p>11.4) $(sec ,x)’ = sec ,xtan ,x$<br>
<code>mathematica     $(sec \,x)' = sec \,xtan \,x$     </code></p>
<p>11.5) $(a^x)’ = a^xln ,x $<br>
<code>mathematica     $(a^x)' = a^xln \,x $     </code></p>
<p>11.6) $(x^{\mu}) = \mu x^{\mu-1}$<br>
<code>mathematica     $(x^{\mu}) = \mu x^{\mu-1}$     </code></p>
<p>11.7) $(cos ,x)’ = -sin ,x$<br>
<code>mathematica     $(cos \,x)' = -sin \,x$     </code></p>
<p>11.8) $(cot ,x)’ = -csc^2 ,x$<br>
<code>mathematica     $(cot \,x)' = -csc^2 \,x$     </code></p>
<p>11.9) $(csc ,x)’ = -csc ,xcot ,x$<br>
<code>mathematica     $(csc \,x)' = -csc \,xcot \,x$     </code></p>
<p>11.10) $(e^x)’ = e^x$<br>
<code>mathematica     $(e^x)' = e^x$     </code></p>
<p>11.11) $$(log_ax)’ = \frac{1}{xln ,a}$$<br>
<code>mathematica     $$(log_ax)' = \frac{1}{xln \,a}$$     </code></p>
<p>11.12) $$(ln ,x)’ = \frac{1}{x} $$<br>
<code>mathematica     $$(ln \,x)' = \frac{1}{x} $$     </code></p>
<p>11.13) $(arcsin ,x)’ = \frac{1}{\sqrt{1-x^2}}$<br>
<code>mathematica     $(arcsin \,x)' = \frac{1}{\sqrt{1-x^2}}$     </code></p>
<p>11.14) $(arccos ,x)’ = -\frac{1}{\sqrt{1-x^2}}$<br>
<code>mathematica     $(arccos \,x)' = -\frac{1}{\sqrt{1-x^2}}$     </code></p>
<p>11.15) $(arctan ,x)’ = \frac{1}{1+x^2}$<br>
<code>mathematica     $(arctan \,x)' = \frac{1}{1+x^2}$     </code></p>
<p>11.16) $(arccot ,x)’ = -\frac{1}{1+x^2}$<br>
<code>mathematica     $(arccot \,x)' = -\frac{1}{1+x^2}$     </code></p>
<ol start="12">
<li>
<h2 id="重要的极限"><a class="header-anchor" href="#重要的极限"></a>重要的极限</h2>
</li>
</ol>
<p>12.1) $\lim_{x\to 0} ,{\frac{\sin x}{x}} = 1$</p>
<pre><code class="language-mathematica">$\lim_{x\to 0} \,{\frac{\sin x}{x}} = 1$
</code></pre>
<p>12.2) $\lim_{x\to \infty} ,{({1+ \frac{1}{x}})}^x =  e$</p>
<pre><code class="language-mathematica">$\lim_{x\to \infty} \,{({1+ \frac{1}{x}})}^x =  e$
</code></pre>
<h2 id="markdown-代码集"><a class="header-anchor" href="#markdown-代码集"></a>markdown 代码集</h2>
<pre><code class="language-mathematica">1. 平方差公式
$ a^2-b^2= (a+b)(a-b) $
2. 立方差公式
$ a^3-b^3 = (a+b)(a^2-ab+b^2) $
3. 立方和公式
$ a^3+b^3 = (a-b)(a^2+ab+b^2)$
4. 完全平方公式
$(a + b)^2  = a^2 + 2ab+b^2 $
$(a - b)^2  = a^2 - 2ab+b^2 $
5. 一元二次方程求根公式
$ ax^2+bx+c =0$
$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
6. 韦达定理
　　设$x1,x2$是一元二次方程$ ax^2+bx+c =0$的两个根，则$x1,x2$满足:
$x1+x2 = -\frac{b}{a}$
$ x1 \cdot x2 = \frac{c}{a}$
7. 有关集合的公式
　　设$I$为全集，$\varnothing$为空集，如果$A$是$I$的子集,$B$是$I$的子集，则$A \subset I $, $ B \subset I $。
于是，
$I \cup A = I$
$I \cap \varnothing = \varnothing$
若$ \bar A =\{x|x \in I且x \notin A,A \subseteq I \}$, 
　$ \bar B =\{x|x \in I且x \notin B,B \subseteq I \}$
则
$ A \cup  \bar A = I$
$ A \cap \bar A = \varnothing$
$\overline {A \cap B} = \bar A \cup \bar B$
$\overline {A \cup B} = \bar A \cap \bar B$

8. 不等式
8.1) $|a| \ge 0$
8.2) $|a|-|b| \le |a+b| \le |a|+|b|$
8.3) $|a| \le b $
8.5) $|a| \le b \Leftrightarrow -b \le a \le b　 (b&gt;0)$
8.6) $a^2+b^2 \gt 2ab　(a,b \in R)$
8.7) $\frac{a+b}{2}  \ge \sqrt{ab}　(a,b \in R^+)$
8.7) $\frac{b}{a} + \frac{a}{b} \ge 2　(ab&gt;0)$
8.7) $\frac{a+b+c}{3} \ge \sqrt[3]{abc}　(a,b,c \in R)$
8.7) $\frac{a_1+a_2+ \ldots +a_n}{n} \ge \sqrt[n]{a_1a_2\ldots a_n}　a_1,a_2, \ldots,a_n \in R^+ n \in N且n&gt;1$
9. 排列组合
9.1) $ A_n^m = \frac{n!}{(n-m)!} $
9.2) $ C_n^m = \frac{n!}{m!(n-m)!} $
9.3) $ C_n^m = C_n^(n-m)$
9.4) $ C_{n+1}^m = C_n^m + C_n^{m-1}$
9.5) $C_n^0+C_n^1+\ldots +C_n^n = 2^n$

10. 积分公式
10.1) $\int sin(x)dx = cos(x) +C$
10.2) $\int cos(x)dx = -sin(x) +C$
10.3) $\int tan(x)dx = -ln|cos(x)| +C$
10.4) $\int cot(s)dx = ln|sin(x)| +C$
10.5) $\int sec^2(x)dx = tan(x) +C$
11. 导数公式
11.1) $(C)'=0 $
11.2) $(sin \,x)' = cos \,x$
11.3) $(tan \,x)' = sec^2 \,x$
11.4) $(sec \,x)' = sec \,xtan \,x$
11.5) $(a^x)' = a^xln \,x $
11.6) $(x^{\mu}) = \mu x^{\mu-1}$
11.7) $(cos \,x)' = -sin \,x$
11.8) $(cot \,x)' = -csc^2 \,x$
11.9) $(csc \,x)' = -csc \,xcot \,x$
11.10) $(e^x)' = e^x$
11.11) $$(log_ax)' = \frac{1}{xln \,a}$$
11.12) $$(ln \,x)' = \frac{1}{x} $$
11.13) $$(arcsin \,x)' = \frac{1}{\sqrt{1-x^2}}$$
11.14) $$(arccos \,x)' = -\frac{1}{\sqrt{1-x^2}}$$
11.15) $$(arctan \,x)' = \frac{1}{1+x^2}$$
11.16) $$(arccot \,x)' = -\frac{1}{1+x^2}$$
12. 重要的极限
12.1) $$\lim_{x\to 0} \,{\frac{\sin x}{x}} = 1$$
12.2) $$\lim_{x\to \infty} \,{({1+ \frac{1}{x}})}^x =  e$$
</code></pre>
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